Lq(Lp)-theory of stochastic differential equations

Pengcheng Xia, Longjie Xie, Xicheng Zhang and Guohuan Zhao

Stochastic Processes and their Applications, 2020, vol. 130, issue 8, 5188-5211

Abstract: In this paper we show the weak differentiability of the unique strong solution with respect to the starting point x as well as Bismut–Elworthy–Li’s derivative formula for the following stochastic differential equation in Rd: dXt=b(t,Xt)dt+σ(t,Xt)dWt,X0=x, where σ is bounded, uniformly continuous and nondegenerate, b∈L˜q1p1 and ∇σ∈L˜q2p2 for some pi,qi∈[2,∞) with dpi+2qi<1, i=1,2, where L˜qipi,i=1,2 are some localized spaces of Lqi(R+;Lpi(Rd)). Moreover, in the endpoint case b∈L˜∞d;uni⊂L˜∞d, we also show the weak well-posedness.

Keywords: Krylov’s estimate, Lq(Lp)-estimates; Zvonkin’s transformation; duality (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (5)

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DOI: 10.1016/j.spa.2020.03.004

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